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Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black-white lattice. For each one of these equations, two different three-leg forms are constructed, leading to…
We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times2$ Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems…
In this paper, we investigate a general quasilinear elliptic and singular system. By monotonicity methods, we give some existence and uniqueness results. Next, we give some applications to biological models.
Some recent developments in the study of exactly solved lattice models in statistical mechanics are briefly reviewed. These include work with Debbie Bennett-Wood and Aleks Owczarek on polymers at surfaces (cond-mat/9805148) and with…
A kind of Bargmann symmetry constraints involving Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable…
We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and…
We investigate the presence of static solutions in models described by real scalar field in two-dimensional spacetime. After taking advantage of a procedure introduced sometime ago, we solve intricate nonlinear ordinary differential…
We prove that multi-soliton solutions of the Toda lattice are both linearly and nonlinearly stable. Our proof uses neither the inverse spectral method nor the Lax pair of the model but instead studies the linearization of the B\"acklund}…
In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases.
A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…
A multilinear M-dimensional generalization of Lax pairs is introduced and its explicit form is given for the recently discovered class of time-harmonic, integrable, hypersurface motions.
In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this…
We propose a conjecture regarding the integrally closedness of lattice polytopes with large lattice lengths. We demonstrate that a lattice simplex in dimension 3 (resp. 4) with lattice length of at least 2 (resp. 3 and no edge has lattice…
We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its…
If $L$ is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex $\Delta(L)$ (definition recalled). Lattice-theoretically, the resulting object is a subdirect…
We consider the problem of solving a linear system of equations which involves complex variables and their conjugates. We characterize when it reduces to a complex linear system, that is, a system involving only complex variables (and not…
In part I: We find a series physical scales such as 1) Planck scale, 2) Minimal approximate grand unification SU(5), 3) the mass scale of the see saw model right handed or Majorana neutrinoes, some invented scale with many scalar bosons,…
The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of…
A new method is proposed to generate nonlinear integrable systems by starting with existing Lax pair and a new form of Kr\"onecker product. It is observed that such equation can be generated with the help of a Hamiltonian structure.…